1907-1993 Zarko Mijajlovi¶c

Kurepa’s early daysand education

Academic careerAdministrativefunctions andmemberships inacademiesOrganization ofscientific work

Awards and honors

Scientific output

LecturesKurepa’s influenceon Yugoslavmathematicians

Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade

Kurepa’s doctoralstudentsKurepa and othermathematiciansKurepa’s work intopology

Kurepa’s work intopology

Kurepa’s work in settheory






Kurepa’s work innumber theory


Conclusion

1 / 25

-DURO KUREPA

1907-1993

Zarko MijajlovicUniversity of BelgradeFaculty of [email protected]

Kurepa’s early days and education



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

2 / 25

Djuro Kurepa was born on August 16, 1907 in Majske Poljanenear Glina in Srpska Krajina as the fourteenth and last child ofRade and Andelija Kurepa. He went to elementary and highschool in Majske Poljane, Glina and Krizevci. Then, he got hisdiploma in theoretical mathematics and physics at the Faculty ofPhilosophy of the University of Zagreb in 1931.

Kurepa spent the years 1932–1935 in Paris at the Faculte desSciences and the College de France. There he obtained hisdoctoral diploma at the Sorbonne in 1935 before a committeewhose members were Paul Montel (president), Maurice Frechet(supervisor), and Arnaud Denjoy.

He was on post-doctoral studies at the University of Warsaw andthe University of Paris (1937). After the Second World War hevisited Cambridge (Massachusetts), the mathematicaldepartments of the Universities of Chicago, Berkeley and LosAngeles, and the Institute of Advanced Studies in Princeton.

Academic career



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Kurepa’s first employment was at the University of Zagreb in1931, as an assistant in mathematics. He became an assistantprofessor at the same institution in 1937, associate professor in1938, and full professor in 1948.

He stayed in Zagreb until 1965 when he moved to Belgradewhere he was invited to be full professor at the Faculty ofScience of the University of Belgrade. He stayed there until hisretirement in 1977. Meanwhile, he was a visiting professor atColumbia University in New York (Summer School 1959), andBoulder, Colorado, in 1960.

Kurepa died in 1993 in Belgrade in the age of 86.

Administrative functions and memberships inacademies



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Besides his university teaching, Kurepa organized successfullyscientific work, as well, and was very active in administrativematters. Professor Kurepa was the chairman of theMathematical Department of the Faculty of Philosophy inZagreb; then since 1970 till 1980 chairman of the MathematicalSeminar of the Institute of Mathematics of SANU (the SerbianAcademy of Sciences and Arts).

He was a full member of SANU, the Academy of Science ofBosnia and Herzegovina, and a corresponding member of JAZU(Yugoslav Academy of Sciences and Arts) in Zagreb.

Organization of scientific work



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Professor Kurepa was the founder and president of the Society ofMathematicians and Physicists of Croatia, and president of theUnion of Yugoslav Societies of Mathematicians, Physicists andAstronomers. He was also president of the Yugoslav NationalCommittee for Mathematics, as well as president of the BalkanMathematical Society.

Furthermore, he was the founder and for many years the chiefeditor of the journal Mathematica Balkanica , now published inSofia. Kurepa was also a member of the editorial board ofBelgrade’s mathematical journals Publications de l’InstitutMathematique, Vesnik and the German journal Zeitschrift furmathematische Logik und Grundlagen der Mathematik.

Awards and honors



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Professor Kurepa earned many awards, honors and distinctions.He received the highest prize of former Yugoslavia, the Award ofAVNOJ (1976).

Also, he was a member of the Tesla Memorial Society of theU.S.A. and Canada (1982), the Bernhard Bolzano Charter, andthe Gramata Marin Drinov of the Bulgarian Academy of Science(Sofia 1987).

Scientific output



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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The scientific output of Djuro Kurepa was rather large. Hepublished about 200 scientific papers, and more than 700writings: books, articles, reviews. His papers were published injournals all around the world, and some of them in the mostrecognized mathematical journals such as:

Matematische Annalen, Izvestiya Akademii nauk SSSR,Fundamenta Mathematicae, Acta Mathematica, ComptesRendus de l’Academie des sciences, Bulletin de la SocieteMathematique de France, Zeitschrift fur mathematische Logikund Grundlagen der Mathematik, Journal of Symbolic Logic,Pacific Journal of Mathematics.

Many of his manuscripts were translated into English, French,Italian, and other languages. Kurepa also wrote about 1000scientific reviews.

Lectures



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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He delivered lectures at many universities of Europe, Americaand Asia; for example, at Warsaw, Paris, Moscow, Jerusalem,Istanbul, Cambridge, Boston, Chicago, Berkeley, Princeton andPeking. As Kurepa himself told once:

I lectured at almost each of nineteen universities of (former)Yugoslavia, then in almost every European country, then inCanada, Cuba, Israel and Iraq, and I gave at least ten lectures ineach of the following countries: France, Italy, Germany, theSoviet Union and the United States.

He participated at dozens of international mathematicalsymposia, and many of them were organized by himself (forexample, the international conferences in topology in HercegNovi 1968, Budva-Becici 1972, Belgrade, 1977.).

Kurepa’s influence on Yugoslav mathematicians



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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The influence of Professor Kurepa on the development ofmathematics in Yugoslavia was great. As a professor of theUniversity of Zagreb he introduced several mathematicaldisciplines, mainly concerning the foundations of mathematicsand set theory. This is best witnessed by the following words ofKajetan Seper, a professor of the University of Zagreb:

Professor Kurepa was not only the professional mathematicianand teacher, but he was a scientist, philosopher and humanist aswell, in the true sense of these words. He was the founder andpioneer in mathematical logic and the foundations ofmathematics in Croatia, and modern mathematical theories inCroatia and Yugoslavia. Generally speaking, he was thecatalyzer, the initiator and the bearer of mathematical science.

From:Set theory. Foundations of mathematics. Symposiumdedicated to Djuro Kurepa. Belgrade 1977

Kurepa in Belgrade



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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His arrival to Belgrade in the mid-sixties, and the subsequentinfluence he had on the mathematical community there, may bedescribed with almost the same words. Professor Kurepaexposed the newest results in diverse mathematical disciplinesthrough many seminars, courses and talks which he delivered atthe Faculty of Sciences and the Mathematical Institute.

The topics of his lectures included: the construction of Cohenforcing, some questions concerning independence results incardinal and ordinal arithmetic, ordered sets and generaltopology. But he was attracted to other mathematical topics,too. He gave valuable contributions to analysis, algebra, numbertheory, and even to those mathematical disciplines which werejust appearing, as computer science, for example.

The universality of his spirit is portrayed by the list of universitycourses he taught: Algebra, Analysis and Topology and SetTheory.

Kurepa in Belgrade



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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By publishing his doctoral dissertation in extenso in PublicationsMathematiques de l’Universite de Belgrade, 4 (1935), 1-138,Kurepa made a first contact with the Belgrade mathematicalcommunity. In the beginning of the fifties these contacts becamedeeper and more frequent. So, from a text

Povodom 35 godina knjige ,,Teorija skupova” profesora DjureKurepe (On the occasion of 35 years of the book ”Set theory” ofDjuro Kurepa), Istorijski spisi iz matematike i mehanike, Istorijamatematickih i mehanickih nauka, knj. 2, Matematicki institut,Beograd, 1989.

of Professor Zlatko Mamuzic we learn that Kurepa was invitedalready in 1952 to visit the University of Belgrade.

Kurepa in Belgrade



Awards and honors

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Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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On this occasion he gave talks on the theory of matrices, andheld a seminar with topics in set theory, topology and algebra. Inthe same text we find the names of attendants of the seminar:Professor Mamuzic himself, Caslav Stanojevic (who later becameProfessor of the University of Rolla, Missouri), Simon Cetkovic(later Professor of the Faculty of Electric Engineering inBelgrade), Mirko Stojakovic (Professor of Algebra of theUniversity of Novi Sad) and others.

By attending these seminars, many mathematicians gained ideasfor their mathematical papers, while graduate students obtainedthemes for their master and doctoral theses. Many of thesethemes were formulated or initiated by Kurepa.

Kurepa’s doctoral students



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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These works include virtually all doctoral theses of the oldergeneration of topologists, and many algebraists from all overYugoslavia:

Svetozar Kurepa, Zlatko Mamuzic, Sibe Mardesic, Pavle Papic,Viktor Sedmak, and a few years later, Ljubomir Ciric, RadeDacic, Milosav Marjanovic, Veselin Peric, Milan Popadic, ErnestStipanic and Mirko Stojakovic. Professor Kurepa was supervisor(altogether 42 times) or a member of examination boards fordoctoral dissertations for many other mathematicians: MiroslavAsic, Natasa Bozovic, Dusan Ciric, Aleksandar Ivic, Milena Jelic,Gojko Kalajdzic, Ljubisa Kocinac, Zarko Mijajlovic, Nada Milicic,Mila Mrsevic, Marica Presic, Zoran Sami, Stevo Todorcevic,Ratko Tosic, and others.

Many of these mathematicians continued and developed furtherKurepa’s work, notably Stevo Todorcevic.

Kurepa and other mathematicians



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Professor Kurepa had contacts with many mathematicians of thehighest rank from all around the world. Thanks to him some ofthem visited Belgrade:

A. Tarski, P. Alexandroff, M. Krasner, N.A. Shanin, K. Devlinand others. Professor Kurepa especially was proud of hisencounter with Nikola Tesla, the great Serbian scientist andengineer, with whom Kurepa was fascinated.

He also met Kurt Godel and Albert Einstein.

Kurepa’s work in topology



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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In topology Kurepa was interested in some generalizations(non-numerical) of distance functions. In this context there is anotion of Kurepa’s pseudo-metric spaces. As it was alreadymentioned, in the middle of thirties Kurepa was a doctoralstudent in Paris, and at that time he was influenced by theFrench mathematical school, particularly by the work of M.Frechet.

Along these lines, Kurepa took a new approach to the notion ofspace. He defined the notion of pseudo-distancial space (espacespseudo-distancies) generalizing the Frechet’s class (D). In thisapproach, values of distance function range over a totallyordered set, instead of the set of positive reals, and the trianglecondition on distance function is replaced by an interestingrelation in ordered sets.

Kurepa’s work in topology



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Later Frechet arrived at the same notion, and since then thisclass of abstract spaces are known as ”Kurepa-Frechet spaces”.A lot of results on this classes of spaces are obtained by Z.Mamuzic, P. Papic, A. Appert, J. Colmez, V.G. Boltjanski andothers.

It is interesting that Kurepa wrote last time about these spaces in19925. Someone probably would recognize in this class of spacesthe notion of Zadeh’s fuzzy sets. Anyway, there is the phrasesets of fuzzy structure”6 in his book Set theory from 1951.

Kurepa’s work in set theory



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Trees, partially ordered sets in which every lower cone is awell-ordered set, may be considered as a natural generalization ofordinal numbers. They are a special type of ramified sets whichKurepa introduced in his doctoral thesis ”Ensembles ordonnes atramifies”. By widespread opinion, for example

K. Kunen, Set Theory, North-Holland, Amsterdam, 1983, p. 69,

this capital work is a first systematic study on set-theoreticaltrees. In his thesis, and later in his papers

Ensembles lineaires et une classe de tableaux ramifies and

A propos d’une generalisation de la notion d’ensamles bienordonnes




Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Kurepa introduced fundamental notions of the theory of infinitetrees:

Aronszajn tree an infinite tree of the cardinality and height ℵ1

in which every chain and every level is at most countable,

Suslin tree, an Aronszajn tree in which every antichain iscountable, and

Kurepa tree, a tree of the cardinality and height ℵ1 which hasat least ℵ2 branches of the height ℵ1.




Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Thus, Kurepa found in Suslin and Kurepa trees two extremalnotions, the first one without long chains (i.e. of the length ℵ1),while the Kurepa tree has a lot of them, at least ℵ2. Kurepaproved many interesting properties concerning these strucure.Probably the best known is the following equivalence with theSuslin hypothesis

SH ⇔ There is no Suslin tree.

Here, SH denotes notable Suslin hypothesis that there is noSuslin line, i.e. a linearly ordered set L of the countable cellularity(i.e. every set of pairwise disjoint intervals of L is at mostcountable), but which does not have a countable dense subset.




Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Lebesgue in 1905 identified implicitly analytic functions with Bairfunctions. In his proof he used an argument which was ”simpleand short”, but it was wrong. The mistaken step in the proofwas hidden in the proof of a lemma which he considered trivial,namely that a projection of a Borel set is also a Borel set.

Ten years later, Suslin, a young and talented Luzin’s studentfound the mistake. Suslin introduced the notion of analytic set,as a projections of Borel sets, and he proved that there areanalytic sets that are not Borel sets. So emerged descriptive settheory, one of the deepest parts of set theory. However, Suslindied soon (in 1919), and the formulation of Suslin hypothesisappear after his death in his paper in FundamentaMathematicae, a year later.

This hypothesis will play the central role in the development ofthe theory of infinite trees, and in this progress Kurepa’s workwas of the principal importance.




Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Kurepa was trying since 1935 to solve SH. He did not succeed inthis, simply it was not possible in this time. Tools of the classicalset theory that Cantor founded, and Zermelo, Fraenkel,Hausdorff, Tarski and others developed, were not adequate.

However, Kurepa was the first who understood the importance oftrees in set theory. Some set-theorists discovered later againproperties of these partially ordered sets. For example, Miller in1943, and Sierpinski in 1948 independently discovered thementioned equivalence which Kurepa found already in 1935.

Using infinite trees, Kurepa found examples of topological spaceswith important and unusual properties. One example of this kindis connected with Suslin line. Namely, Kurepa proved in Lacondition de Suslin et une propriete characteristique des nombresreels, C.R. Acad. Sci. Paris 231, (1950), 1113-1114, that thetopological square of the Suslin continuum K has uncountablecellularity, while K itself has the countable cellularity.




Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Kurepa was not able to prove the existence or nonexistence ofSuslin tree, neither of Kurepa tree. The postulate that there is aKurepa tree was named Kurepa hypothesis, shortly KH. Thecomplete solution of the problem of the existence of these treeswas solved in the beginning of seventies, when the new method,Cohen’s forcing, became a standard and prime tool in set theory.So Solovay, Tennenbaum and Jensen proved that SH isindependent from ZFC + CH (Zermelo-Fraenkel set theory plusContinuum Hypothesis), while K. Devlin proved in 1978 thatevery theory

ZFC ± CH ± SH ± KH

is consistent. This fact shows that the nature of the postulatesSH and KH differs from the others axioms of ZFC. Hence, it isclear why these structures play so important role in variousconstructions in set theory, general topology, model theory andinfinite combinatorics.

Kurepa’s work in number theory



Awards and honors

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Kurepa in Belgrade

Kurepa in Belgrade

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Conclusion

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Kurepa had a distinguished ability to sense a good problem anda fine construction, especially if they are connected to orderedsets. We cannot mention all examples of this kind, but oneproblem from number theory deserves a special attention, as itwas considered by several Yugoslav mathematicians andmathematicians from abroad.

Kurepa formulated during a mathematical gathering in Ohrid in1971 the following problem. First he defined an arithmeticalfunction !n by

!n = 0! + 1! + . . . (n− 1)!

Then the formulation of !n-hypothesis is stated as follows:

The greatest common divisor of !n and n! is 2.

Kurepa’s work in number theory



Awards and honors

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Kurepa in Belgrade

Kurepa in Belgrade

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Conclusion

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This hypothesis has a lot of interesting equivalences, and it wasconsidered by L. Carlitz,Wagstaff, W. Keller, Slavic, Sami,Zizovic, Stankovic, Gogic, Ivic, Mijajlovic, and others.

This hypothesis is stated in R. Guy’s book Unsolved problems innumber theory, Springer-Verlag, 1981, as problem B44.

The hypothesis was tested by computers for n < 1000000(Mijajlovic, Gogic in 1991). Kurepa announced the solution (herang me up one early morning in the spring 1992 to tell me this),but he never published the solution. R. Guy in a letter to me in1991 mentioned that R. Bond from Great Britain might solvedthe left factorial hypothesis, but this proof never appeared.

The !n-hypothesis was finally proved by Daniel Barsky an BenaliBenzaghou in the paper Nombres de Bell et somme defactorielles, Journal de Theorie des Nombres de Bordeaux 16(2004), 117.

Conclusion



Awards and honors

Scientific output


Kurepa in Belgrade

Kurepa in Belgrade

Kurepa in Belgrade











Conclusion

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Kurepa was attracted by many areas of mathematics, besides settheory, general topology, foundations, and number theory. Hiswork includes also topics in algebra (theory of matrices),numerical mathematics, computer science and fixed-point theory.It is not possible to discuss here his full mathematicalachievements. Let us mention that several papers on this matterare already published.

There is also some further discussion in introductory texts of thebook Selected Works of -Duro Kurepa (eds.: S. Todorcevic, Z.Mamuzic, A. Ivic and Z. Mijajlovic). Math. Instit., Belgrade,2005. In short we may say:

-Duro Kurepa has great merits for the development of thefoundations of set theory and mathematics in general.

1907-1993 Zarko Mijajlovi¶c

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